The Calculation
of

Paul Butterfoss
Zachary Tseng
Math 251 – Extra Credit Paper
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Paul Butterfoss
Zachary Tseng
Math 251
Pi () is probably the most well known mathematical symbol. It deals with everything
from trigonometric functions to three-dimensional spheres. One of its most interesting
characteristics is that it is a non-repeating decimal, a totally random series of numbers with no
apparent pattern. Then how is pi calculated? That is what I plan to discus in the following
pages.
One method for calculating pi involves the MacLaurin series of the inverse tangent
function (1).

x 3 x 5 x 7 x 9 x11
x 2 n1




tan-1(x) =  (1) n
= x
……
(1)
3
5
7
9 11
2n  1
n 0
The MacLaurin series for a function (3) is found by taking the Taylor series (2) and
evaluating for the special case a=0.

f ( n ) (a)
f (x) = 
( x  a) n
n
!
n 0
f ' (a)
f '' (a)
f ''' (a)
2
( x  a) +
( x  a) +
( x  a) 3 …….
= f (a) +
(2)
1!
2!
3!

f (x) =

n 0
f ' (0)
f '' (0) 2
f ( n ) (0) n
( x) +
( x) …….
( x) = f (0) +
1!
2!
n!
(3)
When x=1 the MacLaurin series for tan-1(x) is equal to the series expansion for /4. This
is called the Leibniz series (4). (This formula is attributed to Leibniz, although it now seems that
James Gregory (1638-1675) first discovered it).

1 1 1 1
1
12 n1
tan-1(1) =  (1) n
= 1    
……..
(4)
3 5 7 9 11
2n  1
n 0
To compute the number of terms of the Leibniz series needed to reach /4 to a certain
accuracy, we must use the Alternating Series Estimation Theorem (5).
If s = (-1)n-1bn is the sum of an alternating series that satisfies
(a) 0  bn+1  bn
and
(b)
lim b
n 
then
|Rn| = |s – sn|  bn+1
n
0
(5)
In order to reach 6-digit accuracy, bn+1 must be  0.000001. Therefore, the greatest
possible value for bn+1 is 1/1000001 (6).
bn+1 = (1) n
1
12 n 1

2n  1 1000001
(6)
Therefore n must be  500000. Since n=0 is the first term, n=500000 is the 5000001
1
term. But
does not affect the accuracy so there must be exactly 500000 terms in order
1000001
to reach 6-digit accuracy.
Using this series, it takes an enormous amount of terms to simply get 6-digit accuracy.
Therefore we should try to find another series that converges to /4 much faster. One possible
way is through the formula below (7).
4tan-1(1/5) – tan-1(1/239) = /4
(7)
To prove (7), we could use this formula (8) but the coefficient of 4 in front of the inverse
tangent function causes a problem.
x y
tan-1(x) - tan-1(y) = tan-1 (1  xy )
(8)
Therefore we must incorporate the double angle formula for tangent (9).
tan 2x =
2 tan x
1  tan 2 x
(9)
Substituting u = tan x (x = tan-1 u) into this formula we get the following:
2u
)
1 u2
2
tan 1 (
x=
2u
)
1 u2
2
tan 1 (
or
tan-1 u =
2 tan-1 u = tan 1 (
2u
)
1 u2
(10)
(11)
This gives us a coefficient of only 2 however so we need to multiply both sides by 2.
2u
)
(12)
1 u2
Notice however that the right side of equation 12 is the same as the left side of equation
11 with a different value. Therefore we can substitute (2u/1-u2) for u in equation 11.
4 tan-1 u = 2 tan 1 (
Then we get an equation for changing an inverse tangent function with a coefficient of 1
into an inverse tangent function with a coefficient of 4.
2u
2(
)
2
1
1

u
-1
4 tan u = tan (
)
2u 2
1 (
)
1 u2
(13)
Using this equation and plugging in 1/5 for u from the equation for /4 (7), we can get a
value for 4tan-1u without the coefficient of 4. This will allow us to use equation 8 to prove
equation 7.
2(0.2)
)
1  (0.2) 2
1
-1
4 tan (1/5) = tan (
= tan-1 (120/119)
2(0.2) 2
1 (
)
1  (0.2) 2
2(
(14)
Then we can substitute this value into equation 7 to get a new equation that is in the form
of equation 8.
tan-1(120/119) - tan-1(1/239) = /4
(15)
Then using equation 8 and the above values for x and y we can prove that this is true and
that it does equal /4.
120
1

119 239
tan-1 ( 120 1 ) = tan-1(1) = /4
1
*
119 239
(16)
Once we have proven that equation 7 is true, a new series expansion for /4 can be found
by using equation 7 and equation 4.

4
n 0
(1/5) 2n1
2n  1


n 0
(1/239) 2n1
2n  1
= /4
(17)
Again to determine accuracy to a certain number of digits, in this case 6, we must utilize
equation 5. Like in equation 6, bn+1 must be  1/1000001 to achieve 6-digit accuracy. But unlike
equation 4, using this new series expansion we reach 6-digit accuracy much faster. It takes only
4 terms before we have 6-digit accuracy.

4
n 0
(1/5) 2(4)1
2(4)  1


n 0
(1/239) 2(4)1
 1/1000001
2(4)  1
(18)
Since this series expansion for tan-1x allows us to reach higher accuracy much faster, it is
the more efficient of the two series. It is therefore a better series expansion for calculating .
Example: Calculation of Pi to 707-digit accuracy (like William Shanks):
To determine this series to 707-digit accuracy we basically follow the same procedure as
above. We find that we need only 504 terms before we can achieve 707-digit accuracy.

4
n 0
(1/5) 2(504)1
2(504)  1


n 0
(1/239) 2(504)1
 1x10-707
2(504)  1
(19)
The History of the Calculation of Pi
Anyone who knows the simple formula for the area of a circle (Area = pi times the square
of the radius) can compute pi from that relationship (Alfeld 1). However, calculating it
accurately using this method requires much time and patience. Still, this approach to computing
was used for over 3500 years until the late 17th century when the much more efficient calculusbased series expansions became available.
About two thousand years B.C., the first calculations of pi were made, probably by the
Egyptians and Babylonians. A document entitled the Rhind Papyrus, dated 1650 BC, shows that
the Egyptians obtained the value of pi to be (4/3)^4; the Babylonians earlier approximated a
value of 3 1/8 for pi. At about the same time, it is believed that the Indians were using the square
root of 10. These early approximations of pi were not very accurate as they were correct to only
1 decimal place (A Brief History 1).
The next major step towards a more accurate value of pi was taken by the
Greek mathematician Archimedes who decided to take up the problem in about
250 BC. He observed that area of a unit circle (a circle with radius 1) equals the
exact value of pi. By first finding the area of a square inscribed in the circle, then
a pentagon, and so on up through 96-sided polygon, Archimedes continuously
found better and better approximations for pi. With each polygon having more
sides (close to becoming a circle), he came closer and closer to finding an exact value of pi. He
finally discovered that 3 10/71 < pi < 3 10/70. Many people often use the latter value, 22/7,
when dealing with pi (when it does not require precise calculations). In fact, it is used so often
that some people actually think that it is the exact value of pi (A Brief History 1).
Following in Archimedes’ footsteps in 150 AD was Ptolemy of Alexandria. He
estimated the value of pi to be 377/120. The Chinese were a part of this pi race also. In 500 AD
Tsi Ch’ung-Chi gave the value of pi at 355/113. These estimations are correct by 3 and 6
decimal places respectively (A Brief History 1).
Other contributors include:
 Al'Khwarizmi (c. 800 )




Al'Kashi (c. 1430)
3.1416
14 places
Viète (1540-1603)
9 places
Roomen (1561-1615)
17 places
Ludolph Van Ceulen (c. 1600) 34 places
(used Archimedes’s method and a -sided polygon)
After Van Ceulen, we encounter the method that we used to solve the problem in the
previous pages. Wrongly entitled the Leibniz series, it was actually created by James Gregory.
Using the two formulas above (like we did today), John Machin (c. 1706) calculated 100
decimal digits of . (Carothers 1)
Another interesting mathematical formula for pi was discovered during the European
Renaissance by Wallis (1616-1703).
Here the two products continue indefinitely and they finally converge to
the value of pi over 2 (A Brief History 1).
William Shanks (c. 1807) determined the first 707 digits of . This amazing feat took him
over 15 years! Sadly however, only 527 of Shanks' digits were correct. Even after publishing
them 3 times, he had mistakes. Each time he corrected errors in the previously published digits,
new ones crept in. Ironically, his first set of calculations proved to be the most accurate
(Carothers 1).
After the invention of desktop calculators, D. F. Ferguson (c. 1947) raised the total to 808
(accurate) decimal digits. In fact, it was Ferguson who discovered the errors in Shanks'
calculations (Carothers 1).
The current record for most pi digits is held by Yasumasa Kanada and Daisuke Takahashi
from the University of Tokyo with 51 billion digits of pi (51,539,600,000 decimal digits to be
precise) (Alfeld 1). Now that’s a lot of pi. 
(The number on the cover is pi to 2,462 decimal places)
Works Cited
Alfeld, Peter. “Archimedes and the Computation of Pi.” 01 April 1999. University of Utah
05 December 2000 <http://www.math.utah.edu/~alfeld/Archimedes/Archimedes.html>.
“A Brief History of Pi.” The Pi Project. Univ. of Texas. 05 December 2000
<http://www.ma.utexas.edu/users/tyilk/PiProj/PiHistory.htm>.
“Calculating Pi.” The Pi Project. Univ. of Texas. 05 December 2000
<http://www.ma.utexas.edu/users/tyilk/PiProj/PiCalc.htm>.
Carothers, Neal. “The Precomputer History of .” 05 December 2000
<http://ernie.bgsu.edu/~carother/pi/Pi2.html>.